Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(5141\)\(\medspace = 53 \cdot 97 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.698538609674161.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | even |
| Determinant: | 1.5141.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{53}, \sqrt{97})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 41x^{6} - 20x^{5} + 704x^{4} + 76x^{3} + 1439x^{2} + 13522x + 21103 \)
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The roots of $f$ are computed in $\Q_{ 47 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 22\cdot 47 + 42\cdot 47^{2} + 22\cdot 47^{3} + 27\cdot 47^{4} + 43\cdot 47^{5} +O(47^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 5 + 28\cdot 47 + 3\cdot 47^{2} + 12\cdot 47^{3} + 44\cdot 47^{4} + 16\cdot 47^{5} +O(47^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 11 + 42\cdot 47 + 46\cdot 47^{2} + 3\cdot 47^{3} + 41\cdot 47^{4} + 41\cdot 47^{5} +O(47^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 14 + 15\cdot 47 + 2\cdot 47^{2} + 26\cdot 47^{3} + 12\cdot 47^{4} + 31\cdot 47^{5} +O(47^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 18 + 8\cdot 47 + 41\cdot 47^{2} + 4\cdot 47^{3} + 43\cdot 47^{4} + 3\cdot 47^{5} +O(47^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 19 + 21\cdot 47 + 10\cdot 47^{2} + 15\cdot 47^{3} + 29\cdot 47^{4} + 4\cdot 47^{5} +O(47^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 37 + 21\cdot 47 + 27\cdot 47^{2} + 45\cdot 47^{3} + 25\cdot 47^{4} + 28\cdot 47^{5} +O(47^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 39 + 28\cdot 47 + 13\cdot 47^{2} + 10\cdot 47^{3} + 11\cdot 47^{4} + 17\cdot 47^{5} +O(47^{6})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,6)(2,4)(3,5)(7,8)$ | $-2$ | ✓ |
| $2$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $0$ | |
| $2$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $0$ | |
| $2$ | $4$ | $(1,8,6,7)(2,3,4,5)$ | $0$ |